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Table of Contents
Intro; Preface; Acknowledgments; Contents; Acronyms; 1 Cubic Fields; 1.1 Introduction; 1.2 Discriminants and Orders; 1.3 Integral Bases; 1.4 Cubic Polynomials; 1.5 Newton's Method; 1.6 Types of Cubic Fields; 1.7 Arithmetic in the Ring of Integers; 1.8 The Units of OK; 1.9 Simple Continued Fractions; 2 Cubic Ideals and Lattices; 2.1 Introduction; 2.2 The Ideals of O; 2.3 The Ideals of OK; 2.4 Lattices Over K; 2.5 Some Results Concerning 1-Lattices; 2.6 The Prime Ideals of OK; 2.7 The Analytic Class Number Formula; 3 Binary Cubic Forms; 3.1 Introduction
3.2 Reduction of Binary Cubic Forms of Positive Discriminant3.3 Reduction of Binary Cubic Forms of Negative Discriminant; 3.4 Tools of Translation; 3.5 Classical Composition of Binary Cubic Forms; 3.6 Modern Composition and the Class Group; 4 Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler); 4.1 Introduction; 4.2 The Quadratic Fields Associated with a Cubic Field; 4.3 From Quadratic Generators in OL' to Cubic Fields; 4.4 From Primitive 3-Torsion Ideals of OL' to Cubic Fields; 4.5 From 3-Torsion Ideal Classes of L' to Cubic Fields
5.9 Implications of the Multiplication Polynomials5.10 Some General Examples; 6 The Minima of Forms and Units by Approximation; 6.1 Introduction; 6.2 The Least Positive Absolute Value of a Binary Quadratic Form; 6.3 The Least Positive Absolute Value of an IBCF of Negative Discriminant; 6.4 The Least Positive Absolute Value of an IBCF of Positive Discriminant; 6.5 Integers of K of Small Norm; 6.6 Some Algorithms for Units of Cubic Fields of Negative Discriminant; 6.7 Other Identities on the Three Forms; 6.8 The Hessian as a Function of an Algebraic Integer
7 Voronoi's Theory of Continued Fractions7.1 Introduction; 7.2 Relative Minima and Reduced Lattices; 7.3 Some Finiteness Results for Lattices and Ideals; 7.4 Chains of Relative Minima; 7.5 Intersection of Chains of Relative Minima; 7.6 Purely Periodic Chains of Relative Minima; 7.7 Automorphisms and Purely Periodic Chains; 7.8 Representation of All the Automorphisms of a Reduced Lattice; 7.9 Computing Fundamental Units of Cubic Orders; 8 Relative Minima Adjacent to 1 in a Reduced Lattice; 8.1 Introduction; 8.2 The Lattice of Punctures; 8.3 Computing a Prepared Basis for a 1-Lattice
3.2 Reduction of Binary Cubic Forms of Positive Discriminant3.3 Reduction of Binary Cubic Forms of Negative Discriminant; 3.4 Tools of Translation; 3.5 Classical Composition of Binary Cubic Forms; 3.6 Modern Composition and the Class Group; 4 Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler); 4.1 Introduction; 4.2 The Quadratic Fields Associated with a Cubic Field; 4.3 From Quadratic Generators in OL' to Cubic Fields; 4.4 From Primitive 3-Torsion Ideals of OL' to Cubic Fields; 4.5 From 3-Torsion Ideal Classes of L' to Cubic Fields
5.9 Implications of the Multiplication Polynomials5.10 Some General Examples; 6 The Minima of Forms and Units by Approximation; 6.1 Introduction; 6.2 The Least Positive Absolute Value of a Binary Quadratic Form; 6.3 The Least Positive Absolute Value of an IBCF of Negative Discriminant; 6.4 The Least Positive Absolute Value of an IBCF of Positive Discriminant; 6.5 Integers of K of Small Norm; 6.6 Some Algorithms for Units of Cubic Fields of Negative Discriminant; 6.7 Other Identities on the Three Forms; 6.8 The Hessian as a Function of an Algebraic Integer
7 Voronoi's Theory of Continued Fractions7.1 Introduction; 7.2 Relative Minima and Reduced Lattices; 7.3 Some Finiteness Results for Lattices and Ideals; 7.4 Chains of Relative Minima; 7.5 Intersection of Chains of Relative Minima; 7.6 Purely Periodic Chains of Relative Minima; 7.7 Automorphisms and Purely Periodic Chains; 7.8 Representation of All the Automorphisms of a Reduced Lattice; 7.9 Computing Fundamental Units of Cubic Orders; 8 Relative Minima Adjacent to 1 in a Reduced Lattice; 8.1 Introduction; 8.2 The Lattice of Punctures; 8.3 Computing a Prepared Basis for a 1-Lattice